Can matrilineal second cousins marry?

He is your mother's mother's sister's daughter's son. That means your mother's grandmother is common to both of your
Your Mother's grandmother, your grandmother, your mother and then you
Your Mother's grandmother, Your grandmother's sister, her daughter, the boy
You and he are the 4th generations from your mother's side. So your marriage will not be under Sapinda relation. Hence there will not be any problem legally. You can go ahead.
But generally as per our Hindu religion you and that boy will be considered as brother and sister as you are both the children of cousin sisters. But the relation is very distinct. It all depends on the family culture and practices. You can discuss with your parents and the parents of the boy and take their opinion also.
If you are in love and not worried about the elder's decision, you both can sit and discuss and decide. If both of you are above 18 years there will not be any legal issue in getting married to that boy.

When we look at the Sapinda rule, it is very clearly given that such a marriage cannot be established between the 5th generation from the father's side or the 3rd generation from the mother's side. In your case, you are in the 4th generation from mother's side so you are legally eligible to get married as per Sapinda.
Now after this what matters is the thinking of you and your family as well as those with whom you have to marry and their family because in general, you are already in a relationship which is also important.
Now you should talk to the families about this. Feelings and emotions have their own importance but the permission of the family is also an aspect which we cannot completely deny. I hope you will be able to explain your point to your family and both the families will be happy with this relationship. When relationships progress with the blessings of elders, they become successful and strong.

Thank you mam and sir for your response.Both of your opinions will be duly noted.

So is there any way we can get together? Could you please suggest me.

Basically, the Hindu religion prevents marriages in the same gotra and in blood relation but there is a time limit of generation. Once you cross that time limit of generation it is ok to marry among such relations and permitted as well. Sa-pinda according to the marriem webster dictionary in which" sa" means one and the same and "pinda" means a lump of rice given to ancestors which is given to three nearest ancestors. So by this definition as well you have crossed that border of pinda relation.
Since you have already crossed that limit of three-generation from your mother's side and not fall under sapinda relation so in that way you two can marry.
But to get sure you should try to consult a pandit in your region also since these things vary from region to region. If the pandit approves you can go further to get the approval of your families. And also tell this doubt to a senior family member who you trust will support you so that you can have a clear idea about family opinion as well.

The legal condition in this matter is based on the sapinda relationship and once you are out of the purview of that stipulation then it is obvious that there are no legal restriction which can create problems or inconveniences in future when someone asks for the legal validity of the relationship in case of division of property or other things seen in the cases of separation and divorce. So, let us not go to those extreme cases and confine ourselves to seek solution in the present situation.
In Hindu culture marriages in near and dear relationships are not considered correct though in some communities these restrictions are not so severe and it varies from place to place as our country is very big in its geographical extent as well as cultural backgrounds of various communities within the Hindu people. Hence it is imperative that if the community is in consonance for the relationship there is no problem. But otherwise one has to convince one's parents and unfortunately that is not so easy. The only solution is to take a bold step and go ahead with the relationship and may be one gets stern warning from parents and even ousted from their house but from my knowledge and experience in the matter this turmoil happens for a few year only and then parent simply forgive the children and the family is united back. I have seen so many such cases in the families where strictness was there in such matters but later everything becomes normal.
It all depends on the confidence and boldness of the couple whether they can survive in this world without the support of their parents. It is painful and it hurts also when our families leave us on our own. But if someone wants to go ahead with the love of one's life then nothing matters and only the bold and brave tread through such situations. So, you have to weigh these points and take a decision in the matter. Remember it is a tricky situation - one side is your love and other side are the parents. Whom do you choose? It is a dilemma but every dilemma has a solution and it only requires courage to go ahead.

Thank you all for your valuable responses.

If we consider the norm of the society, both of you definitely fall under the Sapinda relationship and this relationship prohibits to go further for the marriage alliance in most of the societies other than a few ones especially in the south where there is no harm in forging such relationships.
Now let us consider how this relationship is going to affect you both in case you both undertake to go ahead for the final culmination of marriage despite the prevailing constraints of the society. First of all, both of you must think how both of you react if the relationship ends with the final knot of marriage. If both of you are not suffering any guilt of such an alliance, go ahead with the relationship but of course, take the consultation of a lawyer who can guide you how best such a marriage can be effected under the special Hindu Marriage Act.
If everything goes well, take the confidence of your parents apart from the parents of spouse and then finalise the marriage with the permission of the court so that no untoward incident happens at the later stage.
Your marriage bond will last depending upon the mutual trust and love after the alliance. Ensure that you couple should not suffer financial constraints after the marriage so that you face constraints in bringing up issues in relation to the expenses of their education, health and for other parameters.