Nonetheless, I wonder if the Latin words are themselves related (if that is indeed knowable). I suppose this question could only be answered in the positive (the opposite being "we don't know" or perhaps "it's unlikely"), but I am posing it here anyway.
Concubina (concubine) derives from the verb cubare, which means “to lie down, to sleep.” In Latin, just like in English, lying and sleeping are generally innocuous words, but can also refer to sex. Cum aliquo cubare = “to lie/sleep with someone” pretty unequivocally means “to have sex.” A concubina is someone who “lies with a man,” i.e., shares a bed as if in a marriage, but without being married – note that this did not originally refer to an adulterous relationship, just one that was not legitimized by marriage; concubina then came to be used euphemistically to refer to an adulteress. The verb cubare is related to cubitum (elbow) and generally the underlying idea appears to be one of bending, or being in a bent shape (as people may do when they lie down to sleep). It is ultimately traced back to the reconstructed Proto-Indo-European root *keu- , *keu̯ə- , with the meaning “to bend, bulge, cavity.”
Concupiscere (to desire strongly, to demand, to claim), on the other hand, derives from the verb cupere, which pretty much means the same (concupiscere is a so-called inchoative form of the root verb, the -sce- is the tell-tale sign, which is a verb describing the beginning of an action, but that is not really effective in the meaning in this particular case). The verb cupere is considered cognate with Sanskrit kúpyati (“boils up, becomes angry”). The idea is evidently one of being in a state of (emotional) unrest and turmoil.
So while both concubine and concupiscence have a more or less direct relationship with carnal desire, the underlying notions are very different, and there is no indication that they are related.
By the way, the name of the god Cupid is related to “concupiscence.”