Add documentation for the fujitsu_hvac integration (#31192)

* Add documentation for the fujitsu_hvac integration

* Rename the documentation to match the new name of the integration and update the name in the documentation itself

* Integrate suggestions from coderabbit

Co-authored-by: coderabbitai[bot] <136622811+coderabbitai[bot]@users.noreply.github.com>

* Remove quality rating

Co-authored-by: Joost Lekkerkerker <joostlek@outlook.com>

* Fix version

Co-authored-by: Joost Lekkerkerker <joostlek@outlook.com>

* tiny tweaks

* Add Climate section

* Fix grammatical issue.

Co-authored-by: coderabbitai[bot] <136622811+coderabbitai[bot]@users.noreply.github.com>

* move config flow to be right after prereqs

* Update source/_integrations/fujitsu_fglair.markdown

---------

Co-authored-by: coderabbitai[bot] <136622811+coderabbitai[bot]@users.noreply.github.com>
Co-authored-by: Joost Lekkerkerker <joostlek@outlook.com>
Co-authored-by: c0ffeeca7 <38767475+c0ffeeca7@users.noreply.github.com>

source/_integrations/fujitsu_fglair.markdown

@@ -0,0 +1,36 @@
+---
+title: Fujitsu FGLair
+description: Control your Fujitsu heat pump or air conditioner that uses the FGLair app
+ha_category:
+  - Climate
+ha_release: 2024.9
+ha_domain: fujitsu_fglair
+ha_integration_type: integration
+ha_codeowners:
+  - '@crevetor'
+ha_config_flow: true
+ha_platforms:
+  - climate
+ha_iot_class: "Cloud Polling"
+---
+
+The {{ page.title }} {% term integration %} provides support for Fujitsu heat pumps and air conditioners that use the FGLair app.
+To find out which app to use for your heat pump, check [the Fujitsu FGLair FAQ](https://www.fujitsu-general.com/global/support/faq/airstage-mobile/0127.html).
+
+## Prerequisites
+
+First, set up your device in the FGLair app before using this integration.
+To configure this integration, you will need the credentials (login and password) used to connect to the FGLair application.
+
+{% include integrations/config_flow.md %}
+
+## Climate
+
+This integration supports the following functionalities (if the devices support them):
+
+- [`set_hvac_mode`](/integrations/climate/#action-climateset_hvac_mode)
+- [`target temperature`](/integrations/climate#action-climateset_temperature)
+- [`turn on/off`](/integrations/climate#action-climateturn_on)
+- [`fan mode`](/integrations/climate#action-climateset_fan_mode)
+- [`swing mode`](/integrations/climate#action-climateset_swing_mode)
+