Michee Efete becomes Brennan's 12th summer signing since taking over from Bobby Wilkinson at the Stones.
The 22-year-old defender joined Billericay from Maidstone in January and scored in a friendly on Friday night.
And speaking to the club's website, Efete was setting his sights high for his new team.
"Getting promotion [is my goal]," he said. "I just want to do as well as I can and help the team - that's my main focus right now and pushing to the [National League].
"In three years time, we could be in League Two; it's possible."
Stones boss Dean Brennan added: "Efete is a player I have worked with before and he will give us plenty of versatility.
"Michee is hungry and leads by example. He is exactly the type of player we want to recruit."
Wealdstone get their 2019/20 National League South season underway against Dartford on August 3.
You can view a full list of Wealdstone's fixtures by clicking here.