Is there a bike route I can take to get from union city, ca to san jose, ca?

If by "bike route", you mean "bike path" (i.e. no cars), then the answer is "NO." There is also not a "direct" bike-accessible road that links those two cities. If you want to bicycle between those two cities, you are going to have to use a combination of bike lanes, bike paths, and streets shared with cars. The answer to your questions also depends on where you are going in San Jose (it's a big city).

I've outlined a route below that will take you from Union City BART to San Jose Airport. I am making the assumption that the person taking this route is an adult cyclist that has experience in traversing urban areas via bicycle. Although this is a fairly leisurely and scenic route, portions of it involve merging with traffic, crossing tricky interchanges, and bicycling along relatively high-speed limited access expressways. I've tried to point out the areas of concern, but there are certainly those that I may have missed. I would not recommend this route for a child, a beginning bicyclist, or at night time. Use it at your own risk.

Disclaimer over... here's the route....

Right on Union Square Drive out of the BART lot. Left on Decoto Rd

Take Decoto RD about 3-4 miles until you cross over highway 880 (the interchange here is a litte tricky for bicycles due to high speeds and merging traffic)

Once over the 880 overpass, there is a bike path that veers off Decoto RD to the right.

Take the bike path alongside Highway 84 until it ends at Lake Blvd (approx 1 mile)

Turn left on Lake Blvd, take it over highway 84, and make the first right onto Jarvis Av

Take Jarvis Av about 2-3 miles - it will curve to the right just over Spruce St and dead end into Gateway Blvd.

Turn left onto Gateway Blvd and go 1 block to Thornton Av

Turn left onto Thornton Av.

Approx 1/4 mile down, you will see the entrance to the Don Edwards SF Bay Nation Wildlife Refuge... make a right to enter the refuge (the name of the entrance road is Marshlands Rd).

Take Marshlands Rd for about 3 miles... it will take you to the base of the Dumbarton Bridge. Cross the bridge using the bike path that runs along the side of the traffic lines.

Once over the bridge, follow the bike path to University Av and turn left. (this is not a great neighborhood, but is definately doable during the daytime, just stay alert in this area).

Take University Av about 2 miles to Donahoe St. Make a left turn (tricky interchange). Donahoe St curves around to the right and becomes E Bayshore Rd (you'll want to follow along the side of the Ikea building).

Take E Bayshore Road for about 5-6 miles through Palo Alto and into Mountain View. After you cross San Antonio Rd (a major street), look for Garcia Av off to your left.

Left turn onto Garcia - which turns into Charleston Rd.

Take Charleston Rd to Shoreline Blvd, make a right.

Take Shoreline Blvd about 1-2 miles to Middlefield Rd, make a left (going over highway 101 on Shoreline involves a tricky interchange with merging traffic; Shoreline/MF is a large multi-lane intersection).

Take Middlefield Rd about a mile and turn right onto Moffett Blvd.

Take Moffet to Central Expressway and make a left (big intersection, be careful).

Take Central Expressway to De La Cruz Blvd. You are now at the northern edge of the San Jose Airport. (Central expressway is a high speed expressway that allows bicycles. It is limited access with on- and off-ramps will require some extra attention on a bicycle).

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In the section between Charleston/Shoreline and Moffett/Central Expressway, there is a bike/pedestrian path called the Stevens Creek Trail. Maybe someone else can provide directions on how to get to it. I cannot remember what the trailheads are, and I don't know what sinage there is on the trail to direct you once you are there.

I don't thik so.