Update API docs for team groups search (#105512)

* Update API docs for team groups search

* update team sync doc

* remove param block

---------

Co-authored-by: Gabriel Mabille <gabriel.mabille@grafana.com>

docs/sources/developers/http_api/team_sync.md

@@ -147,3 +147,54 @@ Status Codes:
 | Action                 | Scope    |
 | ---------------------- | -------- |
 | teams.permissions:read | teams:\* |
+
+**Example Request**:
+
+```http
+GET /api/teams/1/groups/search?name=editors&query=group&page=1&perpage=10 HTTP/1.1
+Accept: application/json
+Content-Type: application/json
+Authorization: Bearer glsa_kcVxDhZtu5ISOZIEt
+```
+
+**Example Response**:
+
+```http
+HTTP/1.1 200
+Content-Type: application/json
+
+```
+
+Status Codes:
+
+- **200** - Ok
+- **400** - Bad Request (invalid team ID format or missing query parameter)
+- **401** - Unauthorized
+- **403** - Permission denied
+- **500** - Internal Server Error
+HTTP/1.1 200
+Content-Type: application/json
+
+[
+  {
+    "totalCount": 1,
+    "teamGroups": [
+      {
+        "orgId": 1,
+        "teamId": 1,
+        "groupId": "cn=editors,ou=groups,dc=grafana,dc=org"
+      }
+    ],
+    "page": 1,
+    "perPage": 10
+  }
+]
+```
+
+Status Codes:
+
+- **200** - Ok
+- **400** - Bad Request (invalid team ID format or missing query parameter)
+- **401** - Unauthorized
+- **403** - Permission denied
+- **500** - Internal Server Error