Blackburn star Blackman rounded off two clinical attacks before forcing home his third goal just before the break to set up a 4-0 SPL win at Fir Park.
The forward took his tally to eight SPL goals from 10 starts on the eve of his 21st birthday and Brown is eager to keep him after his loan deal expires in January.
Brown was delighted to hear Blackburn had also won, at Newcastle, and hopes to capitalise on his colleague's good mood.
"I always phone him (Allardyce) after he wins and tell him he doesn't need another striker," the former Scotland manager said.
"We would be to anxious keep him. I look for the Blackburn result every game because, the more they win, the more chance we have of keeping him.
"He wants to stay, we're hopeful. Sam's been very good with me.
"The deal was arranged in a swimming pool in Qatar in the summer, we were both commentating on the World Cup for Al-Jazeera."
Brown, whose side struck a late fourth through John Sutton, admitted the scoreline had flattered his team.
"In fairness we got every break going," Brown said. "They were excellent goals. Nick's finishing was very good and the creation of the goals was very good.
"That apart St Johnstone were the equal of us and had a lot of pressure, and although Nick correctly will take the credit, I think our goalkeeper (Darren Randolph) deserves a mention because he had a lot of fine saves."
Saints have now conceded 10 goals without reply in three successive SPL defeats, which followed their Co-operative Insurance Cup exit at the hands of Celtic.
Manager Derek McInnes said: "We lost bad goals, I didn't think they had to do a lot to get them.
"There was very little in the game in the first half. We let ourselves down in both boxes.
"We have got to start showing the capacity to defend better and give ourselves an opportunity to win a game.
"I would argue we had more crosses into the box than Motherwell and they scored four goals."