Stock SMS app on Pixel 4a does not have this option and you need alternative methods.
Use SMS Backup and restore
- Option 1 : Take a complete backup of your SMSes and from the app see "Backed up (time)" and tap view details. It will list SMS Backup. Tap on that to view conversations and all your conversations are listed. From the 3 dots menu against the conversation you want select print conversation and it will be printed as a PDF file with date and time stamps. Example below (You can get sent and received. I chose a conversation that doesn’t reveal personal details). From the pdf you can copy and paste wherever needed.
- Option 2: From settings → backup settings → selected conversations only, choose the conversation you want and repeat steps in Option 1 to get a print, share or copy to clipboard.
I am not affiliated to this app but it's likely the best of it's kind.
For those who have MMS messages also and need the same solution , this solution will not work but you have an alternate solution here
Answer from beeshyams on Stack ExchangeI used to have a really good backup application for Android and SMS messages but it was depreciated years ago and quit working within the last year.
I am looking for an application that allows me to backup my text messages and SMS messages. Some things though:
It should allow me too an option to back up a set of specific conversations or everything.
It should maintain the contact name, message date, and contact number for each message.
Messages that are backed up should be displayed in date (conversational) order. I don't want some bullshit where I need to put it together like a puzzle in order to read the conversation.
It should back up all attachments and maintain them in the conversation. If the backup is an HTML document then they can be displayed. If it's an excel document, then clickable links should be inserted into the attachments location within the conversation.
It should allow me to restore any previously saved conversations, including attachments, to my Android phone.
MyPhoneExplorer used to be one of my go tos for this but has since stopped working. I think it was due to a change on the Android OS side. https://www.fjsoft.at/en/
sms - How can I copy entire chain of text messages (not just a single one) with the stock Android messages app? - Android Enthusiasts Stack Exchange
how do i save messages from my phone and put them on to my computer - Google Messages Community
How to Transfer Text Messages from iPhone to Android?
Does Google messages backup text messages?
Videos
If yes then I am unable to backup, if not then any app that can backup text messages?