Mount & Blade II: Bannerlord

In this chicken before the egg scenario, you must find a spouse for your character first to attempt this achievement. This can be done several ways - wait for someone to offer a spouse via political marriage, woo a spouse via conversing with them, or lastly, pay a dowry to a clan leader for one of their spouses' hand in marriage.

Either way, it is probably best doing this as early as you can in your campaign so that you can grow your family tree as soon as possible so that you can have more clan members actively helping your clan grow.

Once married, you may need to go to your clan tab and click the return button (the circular arrow button next to their name) on your spouse to have them travel to your party. Then it is just a matter of waiting for your character & spouse to do the deed. Any time spent waiting gives a chance to produce an heir with your spouse so long as they are together. There is a perk that can up the chances of this as well, fairly early on in the charm skill tree I believe?

From what I can tell, people are getting fairly mixed results on Reddit, some people find this a fairly quick process where they never had to think about it, and others not so much. (I fell in the latter camp).

If RNG is against you, I found making my spouse governor or just a member* of one of my cities and then resting in said city to finally work for me. I also found one tip to be very successful which was to save and re-load that save before attempting to wait, it may be superstition but some people have found loading your save fresh and then waiting, increases chances (this also worked for me after several in-game years of no child while my other clan members seemed to pop out kids straight away).
One thing that I am unsure on if it helped or not, but every in-game week or so, enter a convo with your spouse and trade your entire inventory with them. It instantly gets put back in your inventory but can yield some relationship points which may or may not be necessary but it is something I did in my attempts to get this achievement.

You will get a message once you or your spouse is pregnant and I highly suggest resting your spouse in a castle or city away from any frontlines while the baby is carried to term & not having them in your active party as I have read on Reddit that the baby can die during birth. I haven't played a female character so I am unsure of any possible negative triggers (if any) that could put a baby at risk, such as fighting, injuries etc.

*Thanks KGI KlikoNL for the confirmation.