According to De Vaan's Etymological Dictionary, these two words are not related. Here are his notes on mēns:
PIE *mn-ti- [f.] 'thought, mind'. IE cognates: Skt. mati- [f.] 'thought, mind', Av. *maiti-, Lith. mintis 'thought, idea'
So Proto-Italic and Proto-Indo-European take this mnti form, which has a number of Indo-European cognates (in Sanskrit, Avestan, and Lithuanian, among others).
On the other hand, mēnsa's etymology is different:
PIt. *menssa- 'measured'. It. cognates: U. mefa, mefa [acc.sg.], mefa [abl.sg.], mefe [loc.sg.] 'a certain sacrificial object, maybe cake'.
Here, the cognates are in the Umbrian language. We might not make the connection with "sacrifical objects," but de Vaan indicates that that was originally relevant to the definition – a mēnsa is a "table (for sacred offerings, for meals)."
So no, there's no connection. Mensa International actually employs the "table" meaning anyway, in order to emphasize its nature as a "round-table society, where race, color, creed, national origin, age, politics, educational or social background are irrelevant."