Drew96dawg is correct that any ANDROID PHONE on the Verizon network with 4G LTE service (meaning there is 4G service in the area where the phone is when on the internet and a call comes in)...
and with 4G LTE capability (meaning it takes a SIM card and is subscribed to a data plan that includes 4G LTE service)...
will both surf the internet and make or receive phone calls at the same time.
However, I can not confirm for certain (simply because I haven't tested it), that an ACTIVE 4G LTE internet connection will not be temporarily interrupted while the call is being initiated. In other words, the phone may require a short pause of the 4G LTE Data service in order to initiate an incoming or outgoing call. Once the call is in play, these phones certainly can use the internet simultaneously. It stands to reason that it will maintain that data connection since the LTE radio and the CDMA radio are separate components, but there may be deeper reasons why it can't - if in fact it can't.
Also important to note,
iPhones still do not operate voice and data simultaneously on Verizon and Sprint 4G LTE networks (using the cellular connection), whereas all other 4G LTE phones on their networks can. See;
Sprint and Verizon iPhone 5s have 4G LTE limitations
or
Verizon, Sprint: iPhone 5 can't do voice plus data on 4G LTE | Apple - CNET News
or
https://community.verizonwireless.com/message/912204 (third post from the bottom)
To confirm this, I called Verizon's customer support line at (800) 922-0204, and asked the technician. At first she (as so many others do), shot from the hip and said it does. Then I asked her to confirm as I already knew they didn't. Sure enough...she came back, apologized and said they do NOT work with simultaneous talk and surf on the Verizon network...not the iPhone 4, NOR the iPhone 5.
Now, you can use
Google Voice or
Line 2 or
Skype (or several other VOIP methods), and by phoning with VOIP, you will be able to do simultaneous talk and surf, but it's not CELLULAR talk and DATA SURF. Still, it's not a bad workaround. Besides, VOIP is really the future anyway. Also important to note, if you are on an active WIFI network and have cellular service at the same time, you can make the calls on the cellular network and still surf on the WIFI network. For that matter, even a phone with NO Cellular service can both make calls and surf the web via WIFI simultaneous, so even if the cellular service is poor in your area, if you have WIFI, you can use the methods above to accomplish both.
Regarding why one phone can and another can't make simultaneous calls and surf, the issue here is that the two radios - Cellular and DATA, are separate and independent of each other (in virtually all phones - except iPhones), and so they can operate in tandem. The same holds true to a certain extent with the 3G (1X/EVDO (
EVolution, Data-Optimized)/3G), and 4G radios in Verizon based phones, except that I believe (correct me if I'm wrong), 4G needs 3G to make the initial connection for service, so without a 3G connection, there can be no 4G (at present). This is why when you first power the phone, you'll see 3G making the connection, then shortly thereafter it will change to 4G.
Unfortunately in the iPhones, the 3G/4G radio is the same radio (one radio that can perform either protocol of communications), and so only on HSPA or HSPA+ can negotiate both a cellular call and a data transmission at the same time. This is a design characteristic of the iPhones since Apple originally custom designed them to work on the AT&T HSPA network, and also in an effort to keep costs down (reduced licensing fees and manufacturing costs), and therefore profits up. To combat this and still enable tandem talk and surf, they designed a way to utilize data and Cellular simultaneously with the same radio on HSPA/HSPA+. The iPhones can however, surf and talk if there is a simultaneous WIFI and Cellular (3G or 4G), connection (like like virtually all other phones), since those two radios are separate in any configuration I am aware of.
Only AT&T's 4G HSPA network (which is claimed to be not the same as a true 4G LTE network) supports simultaneous cellular talk and data (not WIFI), surf on the iPhones, AFAIK.