機 in Chinese / Japanese...

機 is an odd one - I've seen this on coffee cups and posters with the meaning of “opportunity.”

機 is a correct definition but this character also means “machine.” In fact, if you put the character for “flying” in front of this character, you have the Chinese word for “airplane” (literally: flying machine). Alone, on a wall scroll, it will be generally understood as “opportunity” but I want you to know this extra information before you make your selection. Note that in Japanese and Korean, this has a similar meaning but can also mean machine or loom.

危機 means crisis in Chinese and Japanese.

Separately, the first character here does mean “danger” or “to endanger,” and the second character can mean “opportunity.”

However, I want to debunk a myth that was propagated by some westerners who did not have a clear understanding of Asian languages...

While often, Chinese/Japanese/Korean compound words (words of two or more characters) are the sum of their parts, this is not always the case. The compound is often understood with a completely different meaning than the two characters individually.

Many have said that the Chinese/Japanese/Korean word for Crisis is made up of the characters for “danger” and “opportunity.” 危機 is true when phrased this way.
However, it's not absolutely correct to say that “danger + opportunity = crisis” in Asian cultures.

English example:
If I tell you that...
Bovine creature + Guy behind the plate in baseball = Locomotive train protection
...you would think I was mad. But consider that “cow + catcher = cowcatcher,” which is the device that used to be found on steam engines to protect them if they hit an animal on the tracks. When we hear the word “cowcatcher,” we don't separate the words into their individual meanings (necessarily).
The same is true with the word for crisis in Chinese/Japanese/Korean. While you can separate the characters, few Asian people would automatically do so in their minds.

The final answer:
It is a half-truth to say, “danger plus opportunity equals crisis” in Chinese/Japanese/Korean. Use this statement and concept with caution.

Also, the second character can mean “secret” or “machine,” depending on context so I guess you have to say “a dangerous machine = crisis” or “danger + a secret = crisis.” Both of these are only slightly more ridiculous than the first premise.

PS: 危機 is probably not a great word for a scroll unless you have a special use for it.

動機 is a common word for motivation in Japanese and Korean.

In Chinese, it can have some bad connotations - so it's safer if your target audience is Japanese or Korean.

機会 is a common way to express “opportunity” in Japanese.

The first character means “chance,” and the second can be translated as “meeting.”

So in Japanese business, a “chance meeting” represents a real “opportunity.”

Note that this also means opportunity in Chinese, but it's more an oral or informal word in Mandarin. Also, the second Kanji is the same as the simplified version of the hui Chinese character.

機遇 is the kind of opportunity that comes via good luck or good fortune.

This word is sometimes translated as “stroke of good luck.”

While there are other ways to express “opportunity,” I think this version is best for a calligraphy wall scroll or portrait.

Note: In Korean Hanja, this would also mean “Meeting someone under strange circumstances.”

時機 is a common way to express day-to-day opportunities.

It's sometimes used to express “an occasion.”

戦機 is a Japanese word that means war machine.

It can also mean the time for battle, military secrets, or battle strategy, depending on the context.

起重機 refers to the huge machine that lifts materials high into the air as crews construct huge buildings.

A customer requested this specifically after some confusion over the bird by the same name.

In an odd twist, while they don't know this name in English sounds like a bird; the building crane is jokingly called “The real national bird of China” because of the accelerated level of construction in Beijing and elsewhere ever since preparations began for the 2008 Olympics. As of 2018, construction has barely slowed.

If you want the type of construction crane that drives down the road, please note that the word is totally different for that kind of “vehicle crane.”

新機軸 is a Japanese word that is used to mean: innovation, new departure, new way, milestone, or breakthrough.

The literal meaning is, “new axis” or “new axle.” It's almost like saying a new wheel (like coming up with a round wheel when the previous version was square).

Please note that this also has meaning in Chinese but only would mean “new axle” without the meaning of “innovation” being associated with it. Consider this to be “Japanese only.”

扳機魚 is the Chinese title for the Trigger Fish.

This literally means “Chance, only once knocks [on the] door.”

機不可失, 時不再來 is often translated as Opportunity Knocks Only Once in Chinese.

The more literal meaning is, “This chance is not to be lost, it won't come again.”