This is an advanced customization that only people comfortable with database management should do. 

To start off, you need to make a new row in your design_elements table. Alternatively you can edit one of the existing button rows like buttonBilling.
 Once you do that, set the fields as this:

did = next highest did available, which is usually 61 on default installs
group_id = button
removable = yes
element_id = buttonGetQuote
description is an optional field, for this case, we won't put anything in it.
caption = Get Quote
type = text
content = Get Quote

Save it or insert it as a new row depending how you created the row.

If everything was set properly, you can now navigate to 'Cart Settings' > 'Appearance Settings' > 'Edit Site Buttons and Text Images' and your new "Get Shipping Quote" field to upload a button should be there.  If it is not, then you need to re examine your mysql entry.

After you upload your button, you will now need to adjust the code in your skin file for the shipping quote page.  To do so, you can either use the file manager , or if you have it available FTP or SSH. 

Navigate to your active skins directory, which by default is /content/skins/flat , and open up shipping_quote.html . 

Comment out the following code using html comments <!-- -->   :

<input type="submit" value="Get Quote"/>

And insert this below it:

<input {$buttonGetQuote}>

Save it, and go to your shipping quote page and see if the button is appearing.  If not, make sure you have set all the code directly or check your error logs as they may provide helpful clues as to how to fix it.