Alker shot a three over par 75 to be in a share of 56th place as he looks to turn around a disappointing year on the second-tier web.com Tour.
Lee and Alker need to finish inside the top 19 players and ties to advance to the final stage.
Meanwhile, at another qualifying event in Florida, Manawatu professional Tim Wilkinson also made a fine start in his attempt to return to the PGA Tour.
Wilkinson carded a four under par 67 to be in a share of ninth place after round one at the Hombre Golf Club in Panama City, only four shots back from Argentine Fabian Gomez.
Wilkinson, who finished 57th on the web.com Tour order of merit, came home strong on the back nine with four birdies in the final eight holes to get right into contention.
Like the Kiwis in California, Wilkinson needs to finish in the top 19 and ties to advance to the final stage.
In Texas, Tauranga professional Josh Geary is another still right in the mix to advance to the final stage.
The 28-year-old, who carded a two under par to sit in a share of 24th place after round one of his qualifying event, backed that up with another 70 to be tied 27th at the halfway stage.
Geary, who has played only five events on the web.com Tour in 2012, is currently one shot off the qualifying number. He needs to finish inside the top 20 players and ties to safely progress to the final stage.