Breaking ciphertext using a quantum computer when the public key is not available As we know about Shor's algorithm on quantum computers it is possible to crack RSA / ECC easily if we have enough qubits. 

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?

Does it matter if we have many ciphertexts and corresponding plaintext for some of them that are generated using the same public key?
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).