As lawful permanent residents of the United States no longer (as of April 26, 2022) need a prior authorization (ETA) before travelling to Canada by air, only a valid green card and a valid passport or travel document are required before boarding the plane (passport is absolutely necessary due to secure flight regulations in Canada), after the arrival of which you may seek entry as permanent resident of Canada with supporting evidences (e.g. expired PR card).
You do not have to present evidence of your status in Canada to the airlines.
However, there have been reports, as recent as July 2023, that some non-Canadian airlines (or their contracted ground staff) are still not familiar with the rule changes in Canada and may insist on an ETA due to outdated information in their system or memory. You have to insist politely but firmly with the evidence from official Canadian government website (https://www.cic.gc.ca/english/helpcentre/answer.asp?qnum=1053&top=16) as well as possibly from the airlines' website (e.g. for United: https://www.united.com/ual/en/us/fly/travel/destination/international.html "U.S. citizens and permanent residents of the U.S. do not need an eTA or a visa.").