I joined company X couple of months back and they sponsored for my
immigration. The offer letter stays that I need to pay the immigration
costs back, if I decide to leave the company with-in 1 year.

During this period, the above mentioned company X is acquired by
company Y and we signed the offer letters from company Y which did not
talk of any of the above mentioned issues related to immigration.

Now, I am planning to move to a company Z. Should I need to pay the
immigration related fees to company X. Are the terms agreed with
Company X during offer are still valid and do I need to abide by the
terms agreed before acquition.

Can I mention  that I agreed to terms for company X and since I am not
comfortable with company Y, I have decided to leave the company
without paying the immigration costs.

Please help me with the above issue or direct me proper resources
where I can get some help.

You might start by asking to inspect your file in the personnel
department and try to discover if and how your obligation the company
x is recorded.
These two sites indicate that under some states' laws, an employee has
a legal right to inspect hir/her file.
http://www.artic.edu/staffresources/eeguide/18-privacy.html
http://www.nolo.com/article.cfm/objectID/5F59F055-9462-49B3-AA3F7925A394E87D/111/259/177/ART/

If the offer letter from company y referred to your contract with x as
a basis for your employment, that would include the agreement about
the sponsoring costs, although not specifically mentioned.  If the
letter only stated that you would be employed in the same position and
at the same salary, but did not included any mention of any agreements
between you and x, you would be in a better position.  But if your
inspection of your file showed that the sponsoring arrangement was
highlighted in some way, obviously you can expect that y will try to
claim the costs.

You  - we -  need to know just what the offer letter from y specifies.

This is, of course, no professional or legal advice, as you can read
in the disclaimer below.

I checked the file and there is no reference to Original Offer letter.

Hi,
Well, that is better than if there were a reference, but no absolute
assurance that the company might not still remember and ask you for
reimbursement.  I expect that it depends on the attitude of the staff
in the personnel department, and they might not care, since it will
cause more work for them, but that is speculation, of course.  If they
like you, maybe agree with your feelings about the company y, it would
be easy for them to "forget" this unusual point in your orignal
contract, and avoid additional work for themselves.
More speculation, but there is probably no "machinery" that would flag
the matter, just human control.

But again, this is no professional or legal advice.

I expect that you and your potential new employee know about the
requirements for employing you as an immigrant, but here is the
information from the US Citizenship and Immigration Service website:
http://www.uscis.gov/graphics/howdoi/h1b.htm

Good luck!  Myoarin