NOC from the present Government office for applying UPSC advertised job

The issue is tricky. When the organisation sends you for any higher studies or specialised training, they may take an undertaking from you. The conditions mentioned in that undertaking are to be followed without any deviation. So you have to see the papers which you have signed before you attend the training programme. There may be a clause saying that after the training the candidate has to work for a minimum of some years. Please check and then decide. If it is not there, what your appointment letter mentions only will be valid.
The notice period if you want to leave the job might have been mentioned in your appointment letter. There may be some other conditions also. You have to see those conditions and then decide.
You have to talk to the authorities who are involved in this aspect and convince them that the job change is important for you. If there is no specific clause. they may give you. It all depends on the recommendations from your immediate boss. You should convince him.
Please keep in mind that changing from one government service to another service is not possible if there is no NOC from the present employer.

You are already in a good government job and now you want to apply for a better government job and you are not getting NOC or permission to apply in your new aspiration. You have mentioned that you were given a special training on account of which your department is telling that you cannot be relieved. Please check the terms and condition of the training and whether you were asked to submit a bond at that time and if yes then find out what is the condition in that bond and if required deposit that amount to get relief from that encumbrance and then apply for the new post.
Meanwhile you have to do one thing that you apply directly in the new post but on that you mention that that it is a direct application. Give a copy of that application to your department and on the top of it you right that this is the through proper channel copy. Please enclose an application to your head of department also that you are applying for such and such post which is also a government post and if selected you will be joining that and in that connection you request them for giving an NOC at the present juncture and then later if you are selected then relieving you from this office.
Please keep copies of all these applications and if your present department doesn't co-operate or help you in the matter then go personally to the higher authorities in your line of control and meet them and show them all the papers that your department is not forwarding them through proper channel.
No department has power to create hindrance or or obstacle in the career growth of a person especially when he is going from a government department to another government department. In any case he will be serving the government only.
I will just remind you that before taking any extreme action first get clear idea about the bond and undertaking which you might had given at the time of going for that specialised training and also ascertain what was the time period for which that bond was applicable or valid.

You need to tackle the matter in a cool way and at this moment, you should apprise of your initiative of applying another government job to the personnel department of your current posting. In that way, you are not suppressing any fact of your further move.
The fate of your application can be of either side - its acceptance or otherwise.
Now you should take all your steps for the preparation of the upcoming interview. If there is anyone in that organisation known to you ask him the pattern of the selection of the candidates for that post. If there is any written interview, what are the patterns of the question set in such an examination. You should prepare accordingly.
In the event of selection, try to apprise of the facts to your present department requesting them to relieve you of the present responsibilities.