Yes, you can connect Vernier sensors to a LEGO NXT or EV3 (the EV3 is LEGO’s newer model robot platform, the NXT is the older platform).

The NXT Sensor Adapter (BTA-NXT) allows Vernier analog (BTA) sensors to be used with the NXT or EV3.

To use the NXT Sensor Adapter with LEGO® MINDSTORMS® NXT software, you need the Vernier NXT Sensor Block. Note that this block is designed specifically for MINDSTORMS NXT and will not work in MINDSTORMS EV3 software. Download the block from here:
https://www.vernier.com/engineering/lego-nxt/vernier-sensor-block/

For a list of sensors supported in the NXT system, scroll down to the NXT Sensor Adapter Compatible Sensors section of this web page:
https://www.vernier.com/products/interfaces/bta-nxt/

The sensor block supports MINDSTORMS NXT version 2.0 and 2.1. Note that this is the Education version. The retail version does not provide data logging or graph viewing. We do not support, or recommend, using MINDSTORMS NXT version 1.1 or 1.0

Vernier Engineering Projects with LEGO® MINDSTORMS® Education NXT contains 12 engineering challenges for use with the LEGO® MINDSTORMS® Education NXT Base Set and LEGO® MINDSTORMS® Education NXT Software version 2. This activity book is a free download and can be found here:
https://www.vernier.com/products/books/ep-nxt/

The NXT Sensor Adapter can also be used in combination with LabVIEW to program NXT and EV3 robots.

The NI LabVIEW Module is a product of National Instruments for NXT/EV3 and includes support for Vernier sensors. Find the download here:
http://sine.ni.com/nips/cds/view/p/lang/en/nid/212785

For additional resources for using the NXT Sensor Adapter with LabVIEW, go here:
https://www.vernier.com/engineering/ni-labview/downloads/bta-nxt/

If you are interested in using Vernier sensors with the LEFO EV3 robotics system, you can find information here:
https://www.vernier.com/engineering/lego-ev3/