No, if you were only a resident of IL for two (2) months (no other iIL income) ,you do not need to file an IL tax return to report $200 of interest income.
To start with, the 1099-INT reports interest income for all of 2017; it is unlikely you received all of it while living in IL. Even if you did, IL does not require Part-year Residents to file a tax return unless they have "earned income" from IL sources. "Interest income" is not "earned income".
From the IL DOR website:
If you were a part-year resident, you must file Form IL-1040 and Schedule NR, Nonresident and Part-Year Resident Computation of Illinois Tax, if
- you earned income from any source while you were a resident,
- you earned income from Illinois sources while you were not a resident, or
