Klein provides a connection in his entry for the root בגד:
Probably denominated from בֶּגֶד (= clothing, garment) and literally meaning ‘to cover with, or as with, a garment’, ‘to conceal’. For sense development compare מעל (= to act unfaithfully, to behave treacherously), which probably derives from מְעִיל (= upper garment, coat); compare also Arab. labisa (= he put on a dress, clothed, dressed), and labasa (= he disguised, he confused), labbasa (= tangle, confusion).
In addition to Klein's mention of meil מעיל - "coat" and me'ilah מעילה - "treachery, embezzlement", I would also add bad בד- "linen" and badah בדה - "to lie, concoct." In fact, English also has that same pairing in fabric and fabricate, and the two meanings of "cloak" (a kind of garment and "to hide, conceal.")
And previously, we've discussed one more: the root חלף - "to change" gives us chalifa חליפה - "change of clothes, suit of clothes", and that verb is also associated with deception (see Bereshit 31:7).
The common thread to all of these is that clothing covers us up, and that cover up can be a source of deception and falsehood. Another theory says that like the "change" of clothes, deceit is considered temporary and unreliable (certainly to the victim), whereas truth is permanent and faithful.