On Sunday, he wrote: "No more vegans, Bhuddists (sic) or Green Party members please."
Martin, 39, is a member of the Green Party and has been a vegan since 2014.
Now, Mr Fraser, who represents Mid Scotland and Fife, has been reported to Police Scotland, The National has reported.
The complainant told the newspaper: "My daughter, who is Buddhist, and many of her community friends feel insulted and marginalised by this statement. Such comments can perpetuate intolerance and division in our society."
Police Scotland confirmed that a complaint had been received.
Previously, a spokesperson for Mr Fraser said the comments were a 'joke', adding: "Russell Martin's period at Rangers perfectly illustrates once again how everything goes wrong when Greens get into positions of power."
The Scottish Greens responded to the jibe yesterday, with a spokesperson saying: "If Rangers are to turn things around then the last person they should be taking lessons from is Murdo Fraser who represents a collapsing party."
Party leader Ross Greer added: "The offensive bit of Fraser's daft tweet wasn't the mention of Greens, it was singling out a religion."
Read more:
'No more Buddhists or vegans at Rangers' says Conservative MSP Murdo Fraser
Letters: Crass Murdo Fraser has shown us why he is unfit to hold public office
Neil Mackay: Murdo Fraser's online trolling sums up the parity of nastiness
Mr Fraser, who has been an MSP since 2001, previously served as Deputy Leader of the Scottish Conservatives between 2005 and 2011.
The party grandee currently serves as Shadow Cabinet Secretary for Business, Economic Growth, and Tourism.