There are a few ways to help combat spam/bots writing into your Messenger.

If the messages are coming from the same geographical location, and it's an area you don't service, you can hide the Messenger from showing to people in those locations. For example if the spam messages are coming from Nigeria and you don't offer services in Nigeria, you can target "Show the Messenger launcher" to "Country is not Nigeria" (for Visitors and/or Users), so it only shows in locations that are not Nigeria:

You can use country, region, city, or continent code there.

Another option would be to enable "Require an email before a conversation starts" in your Automation settings, so that Leads can't start a conversation without first entering an email address. It doesn't completely prevent spam, since a spammer could enter a random email address there, but it does help cut down on bots and less engaged spammers from starting conversations.

Or, if there are certain keywords that the spammers are including in their messages, you can set up a Workflow to automatically close messages that include the keyword:

Just a few options! Let me know if you have any questions.

Hi,
I would suggest that you implement some delays before bots are triggered to reduce instant engagement by spambots. 
Also try restricting or block visibility of bots for visitors from countries where spam activity is prevalent. Further try installing a website security solution like Cloudflare to provide further protection against unwanted visitor bots

@Fario Consulting 

These are great suggestions. Did you know that you can also check if this feature has been recorded on our Product Wishlist in our Community? You can search and upvote existing requests or create new requests Check out the quick guide on how the Product Wishlist works here. Note that you must be logged into the Intercom Community to access the Wishlist. You can log in with SSO or log in as a guest at the top right of the page.

Thanks!