How do I use Google Domain to connect two different web services using a subdomain?

All is about DNS, so the config should be done in your google domain DNS manager if this is your case. [If your Name Servers or NS are pointed from google domains dns manager to another, you should perform the actions in the pointed NameServers DNS management]

Don't modify the root zone records of your example.com, sometimes came with an @ or with the full domainName.com.

So, then create an (A-AAAA)(IP[Hosting Provider{SiteGround site}]) record depending on Ipv4-Ipv6 or a CNAME (example.com) if this is the same as example.com A record for your Siteground routing.

In case of @ just use;

Don't use @, instead use the desired subdomain as in the example

A links xxx.xxx.xxx.xxx TTL(As desired) [if IPv4] or

AAAA links xxxx:xxxx:xxxx:xxxx:xxxx:xxxx:xxxx:xxxx TTL(As desired) [if IPv6] or

CNAME links example.com TTL(As desired) [if Same A or AAA as example.com siteground or other hosting provider]

In case of example.com just use

A links.example.com xxx.xxx.xxx.xxx TTL(As desired) [if IPv4] or

AAAA links.example.com xxxx:xxxx:xxxx:xxxx:xxxx:xxxx:xxxx:xxxx TTL(As desired) [if IPv6]

or

CNAME links.example.com example.com TTL(As desired) [if Same A or AAA as example.com siteground or other hosting provider]

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Remember, here we're just adding, not modifying or deleting any DNS Record.

All of these assuming that the hosting provider is already set up to receive a route from links.example.com which will be the subdomain and also the site to show.