No, they would not be.
At least, to this date[Feb. 2020], there are no known vulnerabilities relating to short messages with sufficiently sized keys. If this is all you cared about, feel free to stop reading here. The rest of the answer just go into details as to why it's not a problem.
According to RFC 2104, a HMAC is defined as follows:
As you can see, the key is XOR'd with a known constant, then concatenated with the message. This data is hashed, then concatenated to other data and finally hashed again. If the message m were just all zero's, it would still not give you any information about the key.
This can be demonstrated by a simpler construct, where all that happens is that the key itself is being hashed. One would not have any information about the key, and the best way of recovering it is brute force.