Every party in the Scottish Parliament has called on Mr Johnson to quit, including Conservative MSPs.
Scottish Tory leader Douglas Ross previously said the Prime Minister's position was no longer tenable after Mr Johnson admitted attending a Downing Street party during lockdown.
However, the party made clear it will not back Mr Cole-Hamilton’s motion and accused the Liberal Democrats of playing “games”.
A spokesman said: “We have made our position very clear on the Prime Minister.
"However, we won’t waste vital parliamentary time on an issue that isn’t within the Scottish Parliament’s remit, particularly when the SNP are failing Scotland on almost every level.
“Mr Cole-Hamilton is using Holyrood to play SNP-style political games instead of focusing on what Parliament should be about – protecting Scottish jobs, restoring Scotland’s schools and improving our public services.
“The Lib Dems are desperately trying to make themselves relevant by taking a page from the nationalist playbook.”
Filing the motion, Mr Cole-Hamilton said: "While new mothers gave birth alone and families said goodbye to their loved ones on Zoom, the Prime Minister and his staff were living it up.
"Boris Johnson isn't sorry that he and his staff repeatedly broke the rules. He is just sad that he got caught.
"The idea that these parties were going on under the noses of Boris Johnson and Rishi Sunak and they weren't aware of it is absolutely farcical. I'm insulted that he thinks the public are so stupid.
"It's time for him to resign as Prime Minister and minister for the union and allow someone else to take over.
"I hope that the Scottish Parliament can speak in one voice in calling for the Prime Minister to go."
The parliamentary motion would have no legal or constitutional effect on Mr Johnson’s position. No date is set for it to be voted on.
The UK Government is reportedly facing 17 separate allegations about parties and other incidents of breaking coronavirus rules, with an investigation currently under way by senior civil servant Sue Gray.