I'm in a religious country, and my religious supervisor is praying. What should I do?

I grew up in a Muslim-majority country. Do's and don'ts:

• Do show respect for the religion. That means avoid doing things like comment on how inconvenient it must be to pray five times a day, don't offer non-halal food, don't invite people to lunch during Ramadan, etc.
• Do do as they do, if possible and not inconvenient. For example if you take lunch together, order/pack something halal. Unless you have special dietary requirements that require you to eat pork, don't eat pork when with others.
• Do dress conservatively. If your institution has a dress code, follow that. If you're female, you can usually refuse to wear the hijab if it makes you uncomfortable (depends on country however), but still dress conservatively - e.g. don't wear something sleeveless, and wear hemlines that go below the knee (this rules out wearing shorts in public).
• Don't pretend to be Muslim. You can't fake it. Learning how to pray in the religion takes time; the chant is nontrivial as well.
• Don't pretend to be Muslim #2: just as important, once you identify as Muslim you could be bound to obey a different set of laws. Sharia criminalizes things which are fine in contemporary Western culture, such as homosexuality. If you're homosexual but identify as Muslim, you could bring the religious police on your head.
• Don't pretend to be Muslim #3: in some countries, if people believe you are Muslim, it can cause you a lot of inconvenience. It doesn't take much to become associated with the religion (extreme historical example, less extreme modern example), and once you become associated, it can be really hard to leave the religion. Potential problems you could face go up to death. In the meantime it's not just you that's affected: in many countries, there are restrictions on Muslim persons marrying a non-Muslim; plus any child born to a Muslim is automatically a Muslim.

Exception: if you were originally a Muslim but no longer believe in the religion, you don't have good options. The above advice will still bring the religious police onto you, because you're now an apostate. You would either have to pretend to be Muslim (at least you should know how to pray) or leave the country and work elsewhere.

tl; dr: Leave religious people to practice their religion and don't criticize them for it. They should do the same to you.

There are situations in which you need to be sensitive to others and they should extend the same courtesy to you. However, there are some places in which religious feeling is so strong as to overcome such sentiment. It can be dangerous in many ways (academic, physical,...) to disagree with religious "consensus" in such places. There are some colleges in the US, in fact, in which this is a problem.

However, to "fake" being a member of a religion is equally dangerous, so you need to be careful. I don't know what people's assumptions about you are. But if they are aware that you aren't a member/adherent of the dominant faith you are probably best advised not to participate. If they are not aware and are making assumptions that you are just an apostate then it can be very dangerous.

If your professor has an open enough attitude you can speak with him/her for advice. But the first rule is to be safe. If you are in a place in which you are required to be a "believer" and you are not, you should work to find a more compatible place. Religious sentiment is often other-than-rational.

I also spent a few years in a MENA country in an academic environment. As you say, eating food/drinking water in public during Ramadan was illegal, as were certain standards of dress, however my experience over the years was that this did not in any way mean that the individuals I lived and worked with (including those who were practicing Muslims) were by default unaccepting of my non-religiousness. I am curious why you say

Surely, if they know I'm "non religious" it will annoy them.

and

So even if my supervisor is kind enough as well as his other doctoral student, but they will be disappointed if he know my situation.

Perhaps you have more reason for believing these than you have included in your question, however, in my personal experience, none of my Muslim friends/co-workers ever expected me to conform to their religion. I certainly never pretended to be Muslim, and like other answers, suggest that you do not do this as this is likely to make the situation more awkward ('I converted to fit in' is probably not the best answer to give to an observant member of any religion).

My advice is that perhaps this does not need to a bigger issue than any other lifestyle difference that you might have with your co-workers, and not to read too deeply into it (as in assuming they have some kind of expectation of you), unless they give you reason for it. Government polices do not necessarily reflect individual expectations. By all means, if you feel unhealthy pressure or feel unsafe, don't stay. Otherwise, similar situations where there is a difference in lifestyle/culture between members of the group occur all the time.

Perhaps another group you join might go out regularly for coffee/alcohol when you are unable to join for dietary/religious reasons, or in another group everyone else has kids, so leaves early, or do play-dates on the weekend. But in such situations, the activity you have to miss out on is surely not the only way to build rapport/respect within your group and strengthen your relationship with your supervisor. So if there is no actual pressure for you to have the same religious beliefs, try to decide if you're bothered by their religious belief. If not, then focus your energy on finding some other topic/hobby that you have in common. If both you and they are respectful of differences, there may be no debate to 'open'.