First if this is a new site, you should definitely switch to Xtreme, as this is the better and supported theme, Weaver II being now end of life.
There is a detailed conversion procedure below to keep what you have so far.
If not the only way to do what you want would be to hide the default Title and tagline (there is an option for that), and insert your own in an HTML insertion area so that you can do what you want to it, like adding the bold around garden (in yellow below)
you could use Advanced options > HTML Insertion areas > Site header Insert code box. Something like
<h1><span><a href=”http://mikeharveygardens.com/” title=”mike harvey gardens” rel=”home”>mike harvey <b>gardens</b> </a></span></h1>
<h2> award winning garden design and planting</h2>
You may need to add inline style to the h1 and or h2 to get the same font styling as the default title and tagline