Does the diameter difference between road and mountain bike handlebars make some bikes incompatible?
This article and Sheldon Brown are good links to look at for the dimensions associated with handlebars. 

In particular, you have to worry about where the stem clamps on, as well as the rest of the bar (where your brake levers &amp; shifters are attached, that is, the grip area) - thus, there are two diameters. The clamp is usually 1 inch for most bikes now (contrary to the first link), but they can differ (older road and MTB bars often use different clamp sizes). Most MTB type components (riser, flat, butterfly, etc.) have a 7/8" rest of the bar while most drop bars (excluding the Walmart ones) are 15/16". So, you can't clamp road shifters/brakes to MTB bars and vice versa. 

To summarize the relevant measurements and their applications, see Sheldon Brown's Handlebar Crib Sheet. 

In this case, the clamp is right (25.4 mm), but the rest of the bar is 7/8". Your shifters are designed for 15/16" bars. This means new shifters (probably thumb shifters or trigger shifters) and brake levers if you switch to this bar, since you can't really bend STI levers to fit. Also, even if you did manage to get them mounted (with some small shimming or something), the shifting action would be pretty unnatural. 

If you want to use trekking bars, I'd probably go with thumb shifters and some short pull brake levers. Note that if you choose to use trigger shifters, they have to be from the Shimano road group (i.e. the Sora flat bar shifters) due to the different cable pull of the road front derailleurs and mountain front derailleurs. Alternatively, you could try a mountain front derailleur with possibly a different crankset (50T/52T rings are big) and use any Shimano mountain bike shifter set, or grip shifts like the SRAM Attack.