This is only an attempt to answer the question, **if** I understood it correctly. You want to *"cover the actual road connection between cities (...) based on the GoogleMap"*.

**First step**: find the coordinates of the two cities you want to use as *edges* in your graphic:

    c1 = CityData["Uppsala", "Coordinates"]
    c2 = CityData["Marsta", "Coordinates"]

> {59.86, 17.64}

> {59.63, 17.84}

**Second step**: define the center of the map:

    center = (c1 + c2)/2

> {59.745, 17.74}

**Third step**: `Import[]` the actual Google Maps image based on the center you've specified

    g1=Import["http://maps.google.com/maps/api/staticmap?center="<>ToString[center[[1]]]<>"," <>ToString[center[[2]]]<>"&zoom=10&size=400x400&sensor=false&maptype=roadmap"]

![enter image description here][1]

I'm still wondering what exactly you want with *"set the calculated distance as EdgeCost value of created edge"*...

**EDITED**

After some research, I think it can also be done:

Define the cities:

    from = ToString[c1[[1]]] <> "," <> ToString[c1[[2]]];
    to = ToString[c2[[1]]] <> "," <> ToString[c2[[2]]];

Now use *Mathematica* to import data from Google with distance and time:

    distdata = Import["http://maps.googleapis.com/maps/api/directions/json?origin=" <> from <> "&destination=" <> to <> "&region=es&sensor=false", {"HTML", "Source"}];

Now find distance by road:

    roadDistance = StringSplit[StringSplit[distdata, ":"][[13]],","][[1]]

> "35.3 km"

Now find travel time:

    travelTime = StringSplit[StringSplit[distdata, ":"][[16]], ","][[1]]

> "30 mins"

**EDITED**

Now put everything together.

First define graphic with markers...

    Import["http://maps.google.com/maps/api/staticmap?center=" <>ToString[center[[1]]] <> "," <> ToString[center[[2]]] <> "&zoom=10&size=400x400&sensor=false&maptype=roadmap&markers=color:blue%7Clabel:O%7C" <> ToString[c1[[1]]] <> "," <> ToString[c1[[2]]] <>"&markers=color:red%7Clabel:D%7C" <> ToString[c2[[1]]] <> "," <> ToString[c2[[2]]]]

![enter image description here][2]

and

    {roadDistance,travelTime}

> {"35.3 km","30 mins"} 


  [1]: https://i.sstatic.net/NszUF.jpg
  [2]: https://i.sstatic.net/7g1FY.jpg