How many NoC can be issued by govt department in a calendar year.

You need not worry much on this regard as practically you may be applying only for such jobs which are better than your current job.

In case you have such doubts, you may consult the HR/ Personnel/staff department in your organisation. Usually no employer will object to the genuine aspirations of better prospects by an employee. However, if the employee is under some service conditions or liabilities, then that have to be complied and satisfied.

Not all recruitments insist on NOC while applying. But they may need the relieving certificate it at the time of joining the new job. So it should be possible for you to resign and get relieved comfortably to join the new job in case you are selected. This and any possible delay due to systems compliance you have to visualise.

Hence, as of now, you need not worry about the number of NOCs etc, but apply for truly and genuinely better opportunities. If an NOC is called o be sent at the first stage of application itself, then you may have to apply for NOC to your employer. Otherwise ensure that you are not under any compulsory service conditions or bond etc and accordingly you can visualise and plan resignation and relief at the time of selection.

As you are working in an Autonomous Body. Now you are thinking of changing the job and apply for other government jobs.
When you apply for a government job, you have to see that all the points mentioned by the recruiter of followed. Otherwise, your application may get cancelled. Some people lost their job after joining also as they were caught for the wrong information they have given in the application.
For many government jobs, you have to apply through your department or you have to attach a NOC. You should follow the instructions given in the notification and accordingly you should apply. Oral information is in no way acceptable and you should have permission for applying in writing.
Generally, if you take a NOC from the employer without mentioning anything about the job you are newly applying for, it will be valid for a period of six months. So once you take a NOC you can apply for different jobs with that NOC during a period of six months. But if they mention the job for which you are applying in NOC that will be useful for that job only. In such case, you have to take for every job. There is no limit mentioned anywhere. But if you go on asking for NOCs many times the present employer may lose interest in having you with him. So you should be careful and should apply for the posts where the chances are very high only.

You need not worry regarding the number of cases of NOC you are requesting your management prior to applying for the posts advertised somewhere else. You need to ensure how best you are fulfilling the criteria of such posts and even your management would not deprive you of such posts for which you are applying.
NOC is a kind of formality to be followed if some employee would look to have a job - switch in the area of his passion but as per service rule, it is essential to be procured so that the management may take up the step of filling up the post either through relocation/ advertising again to fill up the same.
You can straightway resign from the post and join afresh in the newly appointed post but in that way you will be loosing other benefits such as fitment of your basic and your seniority which you could have availed otherwise. Even TA is admissible to the employees applying through proper channel.
However, in some cases, you may not be granted NOC easily in the conditions specified below-
1) If you have filled up a bond of a requisite amount to serve the organisation for a certain period as specified in your appointment letter.
2) You are under the probation- period.
3) Some criminal cases are going against you initiated by your department.
However for the points for the first two points, you may be relieved after paying the dues as decided by the personnel department and NOC can be obtained.
Criminal Cases takes time to get the matter settled and any adverse report in this connection will result in your termination. In that situation, you cannot get a NOC.

There is no stipulation on the number of NOC that an organisation can provide to the employee. When an employee is applying elsewhere, generally the organisation will provide NOC. Only thing is there should not be any condition in the present employment. For example sometimes some organisations which have spent a large amount on the training of a professional will have a bond filled in by the employee before joining the post and in such cases that bond is to be honoured and until one deposits the money as per the condition of the bond, the organisation will not give NOC.

Another important point is that whether the new company is asking for a NOC or not. Generally it is asked to have a confirmation of the credentials of the person. If it is not being asked then one need not to bother for it but at the same time there is one thing that is leaving the old company in time to join the new company and there comes the condition of the employment as how much notice period for leaving the old company is required. If one does not give that time then they are entitled to hold the salaries for a corresponding period which would be a loss to the individual. So in case of NOC not required these things have to be seen in details.