Written Arabic is the same in each country. However, spoken Arabic is different in each country. The dialect of Arabic spoken in Qatar is Qatari Arabic.
Arabic is written from right to left across the page. There are no capital letters in written Arabic. The script is cursive so most of the letters can be joined. Arabic writing is unusual in that the letters can change shape depending on if they come at the beginning, middle or end of the word.
As well as seeing Arabic on shop signs and road signs...
...you can often see Arabic on buildings.
Calligraphy is also be used on works of art.
You can often see geometric and floral patterns in Islamic art. Many examples of this can be seen at the Museum of Islamic Art. Here are some examples of the art found at the museum:
Some of the objects found at the museum are items which were used during battle such as these below: