This was so interesting I googled it!
“Exterminate" comes from "exterminatus," the past participle of "exterminare," meaning "to drive beyond the boundaries." The Latin word "exterminare" was formed from the prefix "ex-" ("out of" or "outside") and "terminus" ("boundary").

Like creativedragonbaby this drove me to googling.
From etymonline com ..
"termination (n.)
late 14c., "authoritative resolution of a matter," from Old French terminacion (13c.) and directly from Latin terminationem (nominative terminatio) "a fixing of boundaries, a bounding, determining," noun of action from past-participle stem of terminare "to mark the end or boundary," from terminus "end, limit" (see terminus). Meaning "end of a person's employment" is recorded from 1961; meaning "artificial end of a pregnancy" is attested from 1969; sense of "assassination" is recorded from 1975."
So the two words have different roots as well as extermination being going beyond termination or limits.
I also found out that the Roman god Terminus was the deity of boundaries, landmarks and the news year celebrations.

Very interesting 8)