No. In God there is only one Father, one Son, and one Holy Ghost because divine perfection and the total absence of matter in God require that there cannot be several Sons in God, etc.
That there are not several Sons, for example, is not due to any lack of begetting power in the Father.
Thus several persons cannot be the term of one notional act; for this four reasons may be given.
The first reason is in regard to the relations by which alone are the Persons distinct. For since the divine Persons are the relations themselves as subsistent, there would not be several Fathers, or several Sons in God, unless there were more than one paternity, or more than one filiation. And this, indeed, would not be possible except owing to a material distinction: since forms of one species are not multiplied except in respect of matter, which is not in God. Wherefore there can be but one subsistent filiation in God: just as there could be but one subsistent whiteness.
The second reason is taken from the manner of the processions. For God understands and wills all things by one simple act. Wherefore there can be but one person proceeding after the manner of word, which person is the Son; and but one person proceeding after the manner of love, which person is the Holy Ghost.
The third reason is taken from the manner in which the persons proceed. For the persons proceed naturally, as we have said (Q41, A2), and nature is determined to one.
The fourth reason is taken from the perfection of the divine persons. For this reason is the Son perfect, that the entire divine filiation is contained in Him, and that there is but one Son. The argument is similar in regard to the other persons.