Fairbrother, the left-handed Lancashire batsman, hit 68 from 75 balls with four boundaries and two sixes in an England total of 214 for six from 38 overs. England then restricted the home team to 170 for seven.
Openers Nick Knight (48) and Vince Wells (31) made sound contributions for England, who then bowled economically to keep the Queenslanders at bay.
Spinners Robert Croft and Ashley Giles were the pick of England's bowlers, Croft finishing with two for 24 and Giles taking two for 23 from their five overs. Martin Love top scored for Queensland with an unbeaten 67.
The Queenslanders had 12 players in their side, in accordance with Australian domestic limited-overs rules, from which 11 bat and field. England chose a traditional 11.
England get the triangular one-day series under way on Sunday with a day-night game against Australia in Brisbane. They then face the Sri Lankans in Brisbane the next day, with eight more games against the two countries to follow.