Short answer. Yes. Family Code Article 22 requires the marriage certificate to state that the parties have entered into a marriage settlement, if any, and to attach a copy of it. A prenuptial agreement is a marriage settlement, so it must be disclosed and physically attached to the certificate.
What the law says
That the parties have entered into marriage settlement, if any, attaching a copy thereof.
Family Code, Article 22 — Contents of the Marriage Certificate. Read the full provision →
What the marriage certificate is required to contain
Article 22 lists what the marriage certificate must state beyond the parties' declaration that they take each other as husband and wife. Alongside items like each party's full name, sex, age, citizenship, religion, and habitual residence, and the date and precise time of the marriage's celebration, the last item on the list is the one that answers your question: that the parties have entered into marriage settlement, if any, attaching a copy thereof. A prenuptial agreement is what the law calls a marriage settlement, so it falls squarely within this requirement.
Disclosure, and physically attaching a copy
The requirement has two parts, and both matter. The certificate has to state that a settlement exists, and it also has to have a copy thereof attached to it. So it is not enough for the certificate to simply note that a prenuptial agreement was signed somewhere; the document itself is meant to accompany the certificate as part of the official record of the marriage, not to exist only in the couple's own private files.
If you have no settlement
The phrase if any makes clear that this requirement is conditional on a settlement actually existing. Couples who marry without any prenuptial agreement are not required to produce one for the certificate; the item simply reflects that no such agreement was entered into. But once a couple has executed a prenuptial agreement before the marriage, Article 22 treats disclosing it and attaching a copy as part of what makes the marriage certificate itself complete, not as an optional extra step a couple can choose to skip.
How this fits alongside the certificate's other contents
The marriage settlement requirement sits alongside the other items Article 22 requires, such as confirmation that a proper marriage license was issued and, where applicable, that parental consent or advice requirements were met. All of these entries exist for the same underlying reason: the marriage certificate is meant to be a complete, self-contained official record of the marriage and the arrangements made around it, not simply a declaration that a wedding took place. Treating the marriage settlement entry as optional would leave that official record incomplete for any couple who did execute one.