No. Only three persons exist in God (the Father, the Son, and the Holy Ghost) because the several persons are the several subsisting relations really distinct from each other.
But a real distinction between the divine relations can come only from relative opposition. Therefore two opposite relations must needs refer to two persons: and if any relations are not opposite they must needs belong to the same person.
Since then paternity and filiation are opposite relations, they belong necessarily to two persons. Therefore the subsisting paternity is the person of the Father; and the subsisting filiation is the person of the Son.
The other two relations are not opposed to each other; therefore these two cannot belong to one person: hence either one of them must belong to both of the aforesaid persons; or one must belong to one person, and the other to the other. Now, procession cannot belong to the Father and the Son, or to either of them; for thus it would follow that the procession of the intellect, which in God is generation, wherefrom paternity and filiation are derived, would issue from the procession of love, whence spiration and procession are derived, if the person generating and the person generated proceeded from the person spirating.
We must frequently admit that spiration belongs to the person of the Father, and to the person of the Son, inasmuch as it has no relative opposition either to paternity or to filiation; and consequently that procession belongs to the other person who is called the person of the Holy Ghost, who proceeds by way of love.
Although there are four relations in God, one of them, spiration, is not separated from the person of the Father and of the Son, but belongs to both; thus, although it is a relation, it is not called a property, because it does not belong to only one person; nor is it a personal relation--i.e. constituting a person.
The three relations--paternity, filiation, and procession--are called personal properties, constituting as it were the persons; for paternity is the person of the Father, filiation is the person of the Son, procession is the person of the Holy Ghost proceeding.