The Q version 3 x 26" is what you want.
I have a Q1, Q2, Q3 in the Garage (err basement, it's winter).
I can confirm the that a 700cx23 will not fit on the Q1 (front and back), Q2 (front no, (back almost)) nor the Q3 x 20" (front)
The Q3 x 26" version will fit a 700cx23 both front and back; the front fork is different on the 26" there is no suspension which is the trick; and the chain stay is longer. If you get a 20" and put the longer chain stay on the 700c tire won't fit in the fork. You must order a 26" version of the Q3 to do that mod. Their are 2 such mods we know of. Steve, Collin.
Depending on the rim design of your wheel set you might be able to get a 25mm or 28mm wheel to fit it varies with the choice of rim (width) and the brand of tire. Example Durano 28 are a lot taller than Contri 28's.
John T has pseudo indorsed the mod "Yes really nice, I might do that mod to mine when I finally take the child seat off the back.".
If you do the full mod you will have left over tires, dd2/3 controls, brake levers, bb5 brakes( if you switch to rim brakes); and a cranks and a bb. Although you can convert the crank in to a nice compact double see the project Quickr thread where I did that. You could source road disc brake wheels which are nice but expensive. If you are paitent you can usually sell them easy on bentrider and ebay; just beware the low ball offers; the DD for example is worth a lot $350-400 and you certain to get a lame old $50 offer at least once.
If you are going to fund such a conversion you might want to seek out some with a Q1 that wants the components you don't need to uplift their rig to internal hubs. That works well we've upgrade a Q1 20" to a Q2x26" equivalent; worked good, and I'm in the middle of uplifting a Q1 20" to a Q2x20" equivalent.
Gotcha's in the project:
1) Not all rim brakes are going to have enough clearance to do the conversion.
2) You need a clamp on cable stop and a clamp on cable pulley to setup the front derailleur.
3) You will need a 40 piece standard bike tool kit
4) A bike maintenance stand makes it a lot easier.
It's a great project if you want to learn about you bike and basic mechanics.
If you wanted some one to do it for you then you have a couple options, Nanda was selling Cruzbikes and has the Skills, Robert certainly could but his new roll might be too busy or less inclined. John Schilter will pretty much build anything if it's possible.
If you tackle it yourself you can get help here in the forums; or you can PM me and I will assist through out the project. Collin would be a good source to pin for a parts list; he was pretty thorough.