I almost quit my job (It implies that I didn't). Ho quasi lasciato il mio lavoro.
Does it mean exactly the same thing, implying that I still have my job?
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In your example the two sentences do have the same meaning (compare the definitions: almost, quasi) but, generally speaking, "almost" and "quasi" are not always equivalent, as "quasi" has one more meaning that "almost" has not.
In fact "quasi" can also mean "as if":
Come, come se fosse ...: [...] Frequente con un part. pass.: avanzava veloce, q. portato dal vento. In funzione di cong. subordinativa (col senso di «come se»), col verbo al congiuntivo: insisteva, q. avesse ragione lui;