A misdemeanor conviction is a criminal conviction, so the honest answer to "Have you ever been convicted of a crime" would be "yes".
I'm not going to promise you what a background check will or will not find; it will depend upon the nature and extent of the background check.
Your state's expungement laws are not the clearest I've ever read, but if these were both misdemeanors it appears that you can petition for expungement. Read this.

