It’s not supported yet, but it seems trivial to add support for it:
// List of Flatpak browsers we support. const QList<BrowserInfo> supportedBrowsers = { {BrowserBase::Firefox, u"org.mozilla.firefox"_s, u"/.mozilla/native-messaging-hosts"_s}, {BrowserBase::Firefox, u"io.gitlab.librewolf-community"_s, u"/.librewolf/native-messaging-hosts"_s}, {BrowserBase::Chrome, u"com.google.Chrome"_s, u"/config/google-chrome/NativeMessagingHosts"_s}, {BrowserBase::Chrome, u"com.google.ChromeDev"_s, u"/config/google-chrome-unstable/NativeMessagingHosts"_s}, {BrowserBase::Chromium, u"org.chromium.Chromium"_s, u"/config/chromium/NativeMessagingHosts"_s}, {BrowserBase::Chromium, u"io.github.ungoogled_software.ungoogled_chromium"_s, u"/config/chromium/NativeMessagingHosts"_s}, };flatpak-integrator/plugin.cpp · master · Plasma / Plasma Browser Integration · GitLab
You can probably copy the files manually as a workaround