Can my unimmunized infant catch chickenpox from my immunized child who has been exposed but has no rash?

Highly doubtful. Infants seldom if ever get chicken pox, because they are protected for the first year of their lives by antibodies received from the mother. That's why children are not given the chicken pox vaccine until age 1. If the mother never had chicken pox or the vaccine, and thus has no antibodies, the baby would be susceptible; but that's a very rare situation these days.
Possibly. Immunization is not a guarantee that someone will not get an infection, though the severity of the infection is often muted or less severe. If your immunized child has been exposed to chickenpox it is possible he/she could expose your non-immunized child. Depending on age of infant, baby may still have protection from mother's immune status gained in utero.