No, HSTS does not protect against certificate misissuance. HSTS simply tells the browser to only allow connecting to that site over HTTPS, it doesn't have anything to do with checking whether the certificate should be trusted.
There are two things that can help with misissuance to some extent, Certificate Transparency (CT) and Certificate Authority Authorization (CAA).
Certificate Transparency won't prevent misissuance by a CA, but it requires that certificates are publicly logged, so you can check and see if any certificates have been issued for your domain that shouldn't have been. Since 2018 Google Chrome has required that all new certificates issued be CT logged in order to be trusted. Firefox does not yet unfortunately, but in practice all certificates issued these days are CT logged as otherwise they would not work with Chrome.
Certificate Authority Authorization allows you to indicate which CAs are allowed to issue certificates for your domain using a DNS record. This will prevent accidental issuance by a CA, but of course a bad actor will just ignore it. Since CAA is only intended to say which CAs are allowed to issue a certificate at a specific point in time, and not for the lifetime of the certificate (ie if the CAA record changes the certificate is still valid), this is only a useful control for CAs that aren't bad actors, as it can help prevent them from issuing a certificate they shouldn't have which could eventually lead to their being distrusted. In fact, the RFC specifically requires that browsers not check CAA records.
A third option is HPKP, which allows you to pin a specific certificate that must be used for your domain (it needn't be a leaf certificate, pinning your CA's root would be somewhat similar to using CAA, but more effective), but it has fallen out of favor and is no longer supported by Chrome.
EDIT: curiousguy points out an interesting attack to work around CT logging in some cases. If an attacker has a Man in the Middle position, they may be able to tell which browser is establishing a connection early on (eg due to minor behavioral differences in the TLS stack). This could allow them to decide not to attack connections from browsers known to check CT logs, simply forwarding the connection to the legitimate server, but present a misissued and non-CT logged certificate to other browsers in order to impersonate the legitimate server. It seems that the only way to prevent this would be for all browsers to check CT logs.