I found an interesting site for comparing names. I can't trace the exact source(s) used for all four names, but the references given for the entire site may prove useful (and impossible to scroll through, admittedly).
- Adam: Originally from the Hebrew אדם, "'adam", meaning "man".
- Eve: Originally from the Hebrew חַוָּה, "chawwah", derived from חוה,"chawah", meaning "to breathe", or חיה, "chayah", meaning "to live".
- Ask: Originally from the Old Norse "askr", meaning "ash tree".
- Embla: Originally from the Old Norse "almr", meaning "elm". The origin of "Embla" is less certain; other scholars give it as "water pot".
Old Norse is, at heart, an Indo-European language. Hebrew, on the other hand, is an Afro-Asiatic language. Given that these two families are extremely different, it appears that their last common ancestral language would have to be millennia ago. Old Norse dates back less than 1,000 years ago, while Hebrew, from its earliest forms, is about 3,000 years old.
There seems to be no connection.