Breaking ciphertext using a quantum computer when the public key is not available
You asked:


  Is it possible to crack RSA / ECC on a quantum computer if we only have ciphertext and don't have the public key used to encrypt itself?


Typically not, with a few exceptions; here are the exceptions:


RSA signatures with deterministic padding (eg. PKCS #1.5) and a small to moderate $e$ (e.g. $e = 65537) - with two signatures (and corresponding message), we can recover the public key with a conventional computer (and then use our quantum computer to break it).
ECDSA signatures - with one signature (and corresponding message), we can recover the public key (with a few "false hits") with a conventional computer; we can then use our quantum computer to break it.


Things that likely look immune:


RSA encryption (which always uses random padding), or randomized RSA signatures (e.g. PSS); the relationships between the plaintexts would appear to be too subtle for the attacker to use them.
EdDSA signatures - because of how the public key is stirred in, the key recovery method we used with ECDSA doesn't work (and without the public key, the attacker can't compute $H(R, A, M)$, which you need to know to have a copy of generating a forgery)
ECIES - the ciphertext consists of $rG$ (for some random $r$) and a symmetric ciphertext that is keyed based of $rP$; the attacker could recover the value $r$ with his handy quantum computer, however without knowing $P$, there's not much else he can do (and having multiple ciphertexts wouldn't appear to help).