Can I take a picture like the one below with the Sigma 18-35 f/1.8?

I own a Nikon D7100

According to the EXIF data it was shot at 10mm focal length on a D7000, so the Sigma you mention wouldn't be quite as wide as this. It was a 4 second exposure at f8, ISO 100.

A wider lens which is excellent quality is the Sigma 10-20mm.

Also, to get the motion blur of the water & clouds, depending on light levels, you MAY need a neutral density filter.

Probably but this is a time (long) exposure that's been edited. Yes you can take the photo but it won't look like this straight out of the camera.

Seeing as the apature on this was cranked down to f8, I wouldn't say the f1.8 is the most important part of this pic. 18mm is a pretty wide lens, not quite as wide as 10mm, but still wide. Given, a wider apature performs better in low light, but that's only if you use it at f1.8. You also lose tremendous depth of field at f1.8, obviously the f8 in this pic was used to capture a wider field depth. If you're looking for landscapes such as this where wide depth of field is the call of the day, focus less on the speed of the lens and more on element quality of the lens. For landscapes I'd rather have an f4.5 with high quality glass in it than an f1.8 cheapie.

Photos such as that one are 90% technique and have little to do with the lens selection.

Other than the increased distortion at 10mm (which likely accounts for the sunburst-like cloud pattern), It matters not if the lens was a 10mm or 18mm, if you are using a 18mm, just back up a bit. The aperture was cranked down to f/8 to obtain a long exposure, as at 10mm virtually everything is in focus anyway as there's an infinite DoF beyond a few feet. Focal length is far more important than aperture than determining DoF

The Sigma 18-35mm f/1.8 is not a cheapie lens, it is probably the best lens for cropped DSLRs made today, and is superior to anything either Canon or Nikon produces.