The pair were given special leave to travel to the capital after their parent died.
Woodhouse told Parliament that after getting assistance while lost, the two women gave their helpers a "kiss and a cuddle".
In her Facebook post, Alkin said the gym member attended a "hands-on" training yesterday morning before she knew her friends had tested positive.
"[She] stayed back for group classes," Alkin posted.
"Fortunate enough we still have the 2m mark down in our class studio, however, my training sessions was hands-on."
Alkin says she has now cancelled her classes and appointments, and decided to self-isolate with her family, until the gym member - who underwent testing this morning - had tested negative.
"I can understand everybody's concern and I would like to ask everyone to please stop private msg me asking for the name of the Lioness Gym member."
The gym member had lent a car to the two returning Kiwis to drive down to Wellington for the funeral.
"Initially the car was dropped to the hotel, however, apparently the 2 girls got lost and my member had to meet with them to give them instructions," Alkin posted.
"[The Ministry of Health] advised me that I can continue with my normal activities until I'll hear back the results of our member," the post says.
Alkin told the Herald she decided to go into self-isolation, despite being told by Ministry of Health officials that she did not have to.
"Imagine if my client is positive and I am positive, and then in the few days until the results come back, there could be hundreds of people involved."
Alkin said she was surprised she wasn't advised to self-isolate, but decided that it would be best to do so.
"I work a lot in the local community, so I decided to self-isolate to be safe," she said.
