Hi Joyce,
I do believe that a dry cough is a side effect (rare) of MTX. I checked
it out on google after reading your post; here's what I found:

GENERIC NAME: methotrexate
BRAND NAMES: Rheumatrex, Trexall
DRUG CLASS AND MECHANISM: Methotrexate is classified as an
antimetabolite drug, which means it is capable of blocking the
metabolism of cells. As a result of this effect, it has been found
helpful in treating certain diseases associated with abnormally rapid
cell growth, such as cancer of the breast and psoriasis. Recently,
methotrexate has been shown to be effective in inducing miscarriage,
for example in patients with ectopic pregnancy. This effect of
methotrexate is attributed to its action of killing the rapidly growing
cells of the placenta. It has also been found very helpful in treating
rheumatoid arthritis, although its mechanism of action in this illness
is not known. It seems to work, in part by altering aspects of immune
function which may play a role in causing rheumatoid arthritis.

PRESCRIPTION: yes

GENERIC AVAILABLE: yes

PREPARATIONS: Injectable: 25mg/ml; Tablet: 2.5mg (Rheumatrex), and 5,
7.5, 10 and 15 mg (Trexall).

STORAGE: Store between 59 and 77degrees F in a sealed container, avoid
light.

PRESCRIBED FOR: Methotrexate is used for cancer treatment generally in
higher doses than for other uses, and is often administered
intravenously or intramuscularly. Methotrexate is used to treat
psoriasis, an inflammatory skin disease, as well as the arthritis that
occurs in 10 percent of these patients (psoriatic arthritis). It is
also used to treat active rheumatoid arthritis in adults and children.
It is also used to treat other rheumatic diseases, including
polymyositis and systemic lupus erythematosus. Methotrexate has been
used to induce miscarriage in patients with ectopic pregnancy.

DOSING: May be taken with or without food. For rheumatoid arthritis and
psoriasis, the dose of methotrexate is given WEEKLY, whether by
injection or orally. For psoriasis, the weekly dose is often divided
into three doses given at 12 hour intervals each week. This has been
shown to be more effective, as it relates to the natural growth cycling
of the skin.

DRUG INTERACTIONS: Because methotrexate can cause serious liver
disease, patients with alcoholism or liver disease should not receive
it. Patients should curtail alcohol consumption while taking
methotrexate. Methotrexate can suppress the body's immunity. Therefore,
any symptoms of infection should be reported to the doctor. Patients
with underlying immune deficiency diseases should not receive
methotrexate. A dry, non-productive cough can be a result of a rare
lung toxicity. Methotrexate can impair fertility, decrease sperm count
and cause menstrual dysfunction. Safety and effectiveness has not been
established in children.

Despite all that, if still could be just a coincidence.
I would definitely call your rheumy and get is opinion.
Good luck to you,
Char