Here are some suggestions that could improve this design:
1- Make the color labels the same width!
This will make it visually more appealing as alignment makes it easier to scan over the page, also sub-consciously offering a calmer reading experience.
2- Use less rounded borders (border radius) for the highlighting color labels to match the same border radius you are using for the Actions icon for unity and consistency, it will also look better overall.
3- Now for the Quotes:
a. I do not recommend moving the quotes to the upper right hand side, because it will end up under the Actions column not the Status! This way it will not be in place and you will be forcing the user to look in different places for the Status.
b. To reduce the clutter, you may represent Quotes by Colors. You are already highlighting the statuses with a colored label. For example: represent the Quote-R by the Green color label, the Quote-* by the Red color label. This is just a suggestion to utilize the colors. I am a little confused by the color labels you are using as I see Query once highlighted in Red and once in Orange. I am not sure if you are utilizing the colors or not based on that.
c. If the colors are already utilized, you may use the letters or symbols without the color labels on the quotes (i.e you are highlighting the R quote with a red color label, use just a the letter R bolded)
d. I noticed a good height for the table's cells. Instead of having the quotes to the right of status text, maybe you can put the them above or below it, depending on which of them you think the user needs to read first the status text or the quote.
e. How about adding a Quotes column?
4- For the Actions column, if you do not have many actions, you may add the actions directly there without the need to click on the dots icon then display the actions then click on the one desired. See this: