Not on the official family terms, which cover a spouse and unmarried children under 21. An adult child needs his own application or another route of his own.
Plan the adult child separately
The official FAQ describes family members as a spouse and unmarried children under 21, each subject to their own processing fee and gift. A 23-year-old does not fit that description, and a parent's permanent residence does not give an adult child any derivative status. His options are separate: if he is studying in the United States he holds his own student status and may later qualify for an employment-based category through a job offer; he could apply for the Gold Card in his own right if he has the means and passes vetting; or, once the parent is a permanent resident, she may file a family-based immigrant petition for an unmarried adult son, which is subject to a preference category with a waiting period. Each has its own cost and timeline. The household should decide whether the parent's plan is worth pursuing if the son's situation is unchanged by it, before paying anything.
Hypothetical example: an adult daughter completing a doctoral program expects to accompany her parent because the parent supports her financially. The first review should compare her age and marital status with the published family definition, then identify whether her own student or employment plans supply a separate route. Her passport, age and the program's current terms settle the derivative question; financial dependence does not change it. Make the household decision before the applicant pays, because the applicant's payment to the U.S. government does not purchase status for an excluded adult child.