Mr Morrison on Monday said the "modern Aus nation" began on January 26, 1788 and that was the day to reflect on what the nation had accomplished, become, and still had to achieve.
"Indulgent self-loathing doesn't make Australia stronger," Mr Morrison tweeted on Monday.
"Being honest about the past does."
Byron Mayor Simon Richardson said the celebrations on January 26 caused pain in a section of the community and questioned whether the values of a fair go and mateship were being reflected.
"Is it true mateship to willingly, willfully and continually to celebrate what rightfully is great to be an Australian on a day that some Australians are pained by?" the Greens representative told 3AW on Monday.
He said the prime minister's response was understandable but he found the remark about "modern Australia" interesting.
"I thought we were actually celebrating Australia Day, not 'modern' Australia Day," he told
"All we're trying to do is trying to reflect history and acknowledge that Australia began, not with the second wave of settlers, but the first."
Mr Richardson's motion was passed at a council meeting last week.