Is a birth certificate enough for a Nexus child application?

It is strongly implied that a birth certificate is sufficient:

Children under 18
When both parents have legal custody, and only one parent or legal guardian is present, a letter of consent is required from the absent parent.

If a parent has sole custody, or if only one parent is recorded on the minor's birth certificate, a consent letter is not required.

If the parents or legal guardian have a custody agreement, it must be provided. The custody agreement must confirm that the child can legally leave Canada.

If a child attends the interview with a person other than their parent or legal guardian, both parents or legal guardians must submit letter(s) of consent in addition to the other requirements.

Minors can be enrolled in the NEXUS program even if their parents or legal guardians are not members.

Specific document requirements can be found here; click on "Step 2 - Gather Documents" for full details:

The following documents are accepted as proof of Canadian or U.S. citizenship/admissibility and permanent residency:

Canadian or U.S. citizen

In Canada a valid passport; a birth certificate and photo ID; a Canadian citizenship certificate or card; or a Certificate of Indian Status.

In the United States a valid passport; a birth certificate and photo ID; a U.S. citizenship certificate; a certificate of naturalization; or a border crossing card.

Canadian or U.S. permanent resident In Canada a valid Permanent Resident Card; or a Record of Landing. In the United States a valid Permanent Resident Card Working or studying in Canada or the United States a work permit; or a study permit.