The modern Verb »atmen« was Middle High German (MHG) »atemen« and Old High German (OHG) (10th century) »atamon«. It is derived from the noun »Atem« (MHG »atem«, OHG »atum«). Etymologist believe, that the closest form in other old languages is Old Indian »atma« or »atman« which means breath, breeze, mind, soul, the self.
The word »Atmosphäre« was brought into German language as a foreign word used by scientists in the 17th century as a composition of the greek words
- ἀτμός (atmós) = steam, vapor, mist, haze, breath, breeze
- σφαῖρα (sphaira) = ball, sphere
The first of the two words is closely related to the verb ἄημι (áēmi) which means "to breath, to blow" and both are derived from Proto Indo European »áwēmi«. This again is also the root of English and German noun »Wind« and from latin »ventus« (which means wind and can be found in German »Ventilator« but also »Ventil«)
In my researches I did not find a clear statement that says, that Old Indian »atma« is related to greek »atmós« (both can mean breath, breeze) or to Proto Indo European »áwēmi«. But since the meaning of these words is so similar, and also the words themselves sound so similar, I believe, that there is a connection. But, as just said: I didn't find any clear statements about such a relationship.