Your booking website is live, but when you search the property name on Google, the property does not appear in the results.
Every property already has its own page. Each property has its own HTML page at /property/<slug>/ on your site. The property name is in the page title, the H1, and the meta description. The page is also listed in the property sitemap published at /sitemap_index.xml, so Google can crawl and index it.
Check whether the page is indexed. Search Google for site:yourdomain.com. If the property page appears in those results, it is indexed.
Why it may not rank yet. Ranking is a separate matter from indexing. A domain that has been live for only a few days will not outrank Airbnb, VRBO, or Booking.com listings that have years of history for the same property name.
Improving ranking over time. You can do three things. First, add the property name and your website address to your OTA listing titles and descriptions. Second, set up a Google Business Profile that references the site. Third, fill in the SEO title and meta description fields in Rank Math on the property page itself. These fields survive PMS syncs.
Inspecting indexing at the URL level. Add your domain in Google Search Console as a Domain property using DNS verification. Your site team can add the required TXT record.