Q: How can backup my photos if I don't remember my iphone passcode

You don't, if it is asking you for your passcode. Have you tried importing the photos instead of trying to backup? While this is a trusted computer, have you ever backed up and/or imported photos to this computer in the past?

If you cannot get into the phone, there is no way to access it. You have to unlock the phone before you can back it up the first time. You have to unlock it to be able to import the photos as well. The only alternative you have is to remember the passcode. If you cannot remember the passcode, the only way to be able to access the device is to restore it as new to remove the passcode.

The passcode is a security feature. Creating a workaround just allows for exploitation and the possibility to steal data. Let me ask you this, how long have you had the phone? How long have the photos been on the phone. iTunes and iCloud provide you several options for backing up your device, and the iCloud one does not require any input from you after the initial setup. You just connect it to power and Wi-Fi, such as going to bed at night, and lock the screen. The device will automatically backup ever day.