The short answer is yes, the HOA can bill you.
If you disagree and are not able to resolve this, most likely the HOA will sue you to collect, in which case, you can present your side of the story to the judge and the HOA will have to present its evidence that you owe whatever it claims you owe.
Frank W. Chen has been licensed to practice law in California since 1988. The information presented here is general in nature and is not intended, nor should be construed, as legal advice for a particular case. This posting does not create any attorney-client relationship with the author. For specific advice about your particular situation, please consult with your own attorney.
You should insist upon being provided with proof that the charges are your responsibility. Did other unit owners also receive the same letter? Can you get a statement from the cleaning service "agent"? You may want to request an "Internal Dispute Resolution" process under California Civil Code section 1363.840. The Board of the HOA is required to designate one of its members to meet with you to try to reach an informal resolution of the dispute.