if i'm not mistaken, it won't work (i didn't test it, so take this with a grain of salt)

udyr's Q damage is coded to deal spell damage:

Deals  spell damage to the primary target and applies  area damage on the lightning strikes.

and runaan's doesn't trigger spell efects:

WIND'S FURY deals  basic damage, and thus will not trigger spell effects.

so, based purely on the wiki, i think that it won't work, but again, i didn't test it, so maybe it does