If the government discovered extraterrestrial life would they tell us?

No.  I saw a Nasa Scientist on TV a year or so ago.  He said a LOT of the Scientists were Religious fundamentalists (of all places) and were against revealing extra terrestrial life if found.   Along with the less religious types who erroneously think any such revelations would disrupt religions and cause chaos in society.

Foolishness to be sure, I am a practicing METHODIST, (and a Scientist) and I think they are full of crap.  Nearly ALL religious people I know have NO PROBLEM with the idea of ET life, even intelligent life.  To think otherwise would be "Limiting God"  the fundies are a small minority of the religious community us "normal" people take little notice of, save for the occasional "Screwball" sects that kiss rattlesnakes or such.

Nasa is even afraid to suggest "Extinct" microbial life on other planets, much less more complex organisms or intelligent life.

Most people have a grade 4 understanding of things 

Religions have repeatedly said God created us solely so any government or religious organization would hide it 

It wouldn't be the government that discovered that life... it would likely be a space agency, or an institution run by universities or private science foundations - like SETI...

The government may try to keep a lid on the discovery, but I think the cat would be out of the bag long before a plan to keep it sealed was developed. 

If you are an American citizen then I have no idea how your government will react, if it is a group of independent scientists, then, possibly yes.

It's unlikely to be the government per se which discovers alien life.  More probable is that it would be a team of researchers working at a university somewhere.  Since the academic community communicates openly, the chances are they'd issue a press release and it would be hard for a lay person to interpret.  For instance, it might be about spectographic analyses of reflected light from one of the moons of Jupiter or the periodic variation in levels of a certain relatively reactive gas in a planetary atmosphere.

I realise you have intelligent life in mind.  There is reason to suppose that the government wouldn't learn about it any earlier than an arbitrary member of the public because a society which survived long enough for members of it to arrive in another solar system would be fully egalitarian and they wouldn't respect or possibly even understand our power hierarchies.

And if they said they had -- you would believe them --and if they said they hadn't , you wouldn't .

Get where I'm going with this ? 

And how would "the government" (which one?) discover extraterrestrial life? It's not as if "the government" had exclusive use of telescopes and radio-telescopes.

Very few American astronomers work for "the government" and most astronomers are not American. For an astronomer, the glory comes from being the first to publish any new discovery. The only concern that we should have is to make sure that whatever discovery we publish has been verified by some other astronomers.

Governments (there are thousands of them) usually "discover" our discoveries the same way people find out: they read the papers (scientific papers, that is).

Remember, back in 2004, when a few astronomers discovered an asteroid (number 99942) and used the few observations to calculate its orbit, finding that it had some probability of hitting Earth in 2029. Then the asteroid's orbit took it behind the Sun where it could not be observed for a few months.  The prudent course would have been to wait for it to be visible again (in early 2005) to confirm the orbit calculations.

But no, they had to publish their results right away in case somebody else had also discovered it. They even gave it a name: Apophis (Ancient Egyptian god of chaos). When "the government" found out, the papers had already been published (in general, governments do not control scientific journals) and the news were all over the press: "We were doomed!"

When observations resumed in 2005, it became clear that Apophis would NOT hit Earth.

If a team of astronomers discovered extraterrestrial life, it would be a discovery so important that it would be published even before other astronomers (never mind governments) heard about it.

Yes, the government hides stuff from us. Top secret military weapons and aircraft, location of submarines, nuclear launch codes. Some information must be kept away from our enemies. But you can not conclude that they are hiding information about alien contact. We have no evidence of that. The government is as clueless about strange things in the sky as we are. 

If you believe that the US government is hiding alien information, why have those aliens only come to the USA and not other countries? If the aliens are all over the earth, do you really believe that 200 countries are all hiding facts about them? Wouldn't a few governments release proof of those aliens? 

Given that astrobiology is such a hot topic now, finding any sort of life (present or past) past Earth would be a monumental discovery.

It would be really hard to keep a lid on such a discovery.

Which government? there are hundreds ...

US, depends on whether or not Trump felt it would help his re-election.