Murray, who lifted the doubles titles at Queen’s alongside Feliciano Lopez last week, and Melo fell to Colombian top seeds Robert Farah and Juan Cabal 6-2 6-4 in the first round.
The three-time Grand Slam champion was broken three times in the defeat, with the rest of the match going on serve.
He will team up with Frenchman Pierre-Hughes Herbert in the men's doubles at Wimbledon next week, but is still seeking a partner in the mixed event.
Shanghai Masters doubles
Colombians Cabal and Farah seal year-end doubles top ranking
09/10/2019 AT 15:09
British duo Dan Evans and Lloyd Glasspool await Farah and Cabal in the quarter-finals.
Murray had hip resurfacing surgery in February in a bid to salvage a career that appeared to be over when he lost an emotional first round match at this year's Australian Open.
The former world number one has reported being "pain free" as he takes his first steps back on Tour in doubles before hopefully resuming his singles career later this year.
He has been moving freely alongside Lopez this week and was clearly fired-up on Sunday as he won his first doubles title since 2011 when he partnered brother Jamie in Tokyo.
Eurosport is the only place to see Jo Konta in action in all three of the grasscourt tournaments leading up to Wimbledon
ATP Queen's doubles
Fist-pumps and tumbling volleys: How Murray returned to action in style
20/06/2019 AT 20:54
ATP Queen's doubles
Murray marks competitive return with doubles victory at Queen’s alongside Lopez
20/06/2019 AT 19:13