Change Page Title
Many plugins and themes might be controlling this area. However, despite this, if you change your Title on the Home Page, it should update your HTML title tag also. This will then update on Google when it next indexes your site, which might take several days.
Google and other search engines use the Title Tag for the title field in search results.
Wordpress will set the Title Tag of your page using the title of your page as seen in the image below.
Yoast Title Change
If you are using the Yoast SEO plugin, then you will also need to check your Yoast settings are correct. Go to the following area and check there.
SEO > Search Appearance > Content Types > Pages
Each snippet grabs the relevant data. So where you see the purple Title bubble in the below image, this is grabbing the page title you set.
Custom Yoast Page Title Settings
*Note: Make sure you haven't overwritten the main setting via custom settings on the actual page via the Yoast tab, as seen in the below image.